Question

If \(x \frac{d y}{d x}+y=\frac{x f(x y)}{f^{\prime}(x y)}\), then \(|f(x y)|\) is equal to
(A) \(\mathrm{ke}^{\mathrm{x}^{2} / 2}\)
(B) \(\mathrm{ke}^{\mathrm{y}^{2} / 2}\)
(C) \(\mathrm{ke}^{x^{2}}\)
(D) \(k e^{y^{2}}\)

Answer 1

Ans: (A)
Hint \(: x \frac{d y}{d x}+y=\frac{x f(x y)}{f^{\prime}(x y)} \Rightarrow \frac{x d y+y d x}{d x}=\frac{x f(x y)}{f^{\prime}(x y)} \Rightarrow \frac{f^{\prime}(x y) d(x y)}{f(x y)}=x d x \Rightarrow \ln f(x y)=\frac{x^{2}}{2}+C \Rightarrow|f(x y)|=k e^{x^{2} / 2}\)